Abstract
For a discrete-time, closed, cyclic queueing network, where the nodes have independent, geometric service times, the equilibrium rate of local progress is determined. Faster nodes are shown to have a capacity depending only on the service probabilities. A family of such networks, each with the same number of types of nodes, is analyzed. If the number of nodes approaches infinity, and if the ratio of jobs to nodes has a positive limit and each node type has an asymptotic density, then for a given node type, the limits of the proportion of occupied nodes and the expected queue length are calculated. These values depend on the service parameter and on the asymptotic rate of local progress. The faster nodes can attain their capacity only when the limiting density of nodes of slowest type is zero.
Similar content being viewed by others
References
N. Bambos, On closed ring networks, J. Appl. Probab. 29 (1992) 975–995.
O. Boxma, F. Kelly and A. Konheim, The product form for sojourn time distributions in cyclic exponential queues, J. Assoc. Comput. Mach. 31 (1984) 128–133.
H. Daduna, The cycle time distribution in a cycle of Bernoulli servers in discrete time, Math. Methods Oper. Res. 44 (1996) 295–332.
H. Daduna and R. Schassberger, Networks of queues in discrete time, Z. Oper. Res. 27 Ser. A (1983) 159–175.
W. Gordon and G. Newell, Closed queueing systems with exponential servers, Oper. Res. 15 (1967) 254–265.
J. Hsu and P. Burke, Behavior of tandem buffers with geometric input and Markovian output, IEEE Trans. Commun. 24 (1976) 358–361.
H. Kobayashi and A. Konheim, Queueing models for communications system analysis, IEEE Trans. Commun. 25 (1977) 2–29.
E. Koenigsberg, Twenty-five years of cyclic queues and closed queue networks: A review, J. Oper. Res. Soc. 33 (1982) 605–619.
M. Miyazawa, Discrete-time Jackson networks with batch movements, in: Stochastic Networks, eds. P. Glasserman, K. Sigman and D. Yao, Lecture Notes in Statistics, Vol. 117 (Springer, New York, 1996) pp. 75–94.
M. Miyazawa and H. Takagi, Editorial introduction: Special issue on advances in discrete-time queues, Queueing Systems 18 (1994) 1–3.
V. Pestien and S. Ramakrishnan, Asymptotic behavior of large discrete-time cyclic queueing networks, Ann. Appl. Probab. 4 (1994) 591–606.
V. Pestien and S. Ramakrishnan, Features of some discrete-time cyclic queueing networks, Queueing Systems 18 (1994) 117–132.
H. Schassberger and H. Daduna, The time for a round trip in a cycle of exponential queues, J. Assoc. Comput. Mach. 30 (1983) 146–150.
V. Sharma, Open queueing networks in discrete time-some limit theorems, Queueing Systems 14 (1993) 159–175.
A.L. Stolyar, Asymptotic behavior of the stationary distribution for a closed queueing system, Problems Inform. Transmission 25 (1989) 321–331.
J. Walrand, A discrete-time queueing network, J. Appl. Probab. 20 (1983) 903–909.
J. Walrand, An Introduction to Queueing Networks (Prentice-Hall, Englewood Cliffs, NJ, 1988).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Pestien, V., Ramakrishnan, S. Queue length and occupancy in discrete-time cyclic networks with several types of nodes. Queueing Systems 31, 327–357 (1999). https://doi.org/10.1023/A:1019122617412
Issue Date:
DOI: https://doi.org/10.1023/A:1019122617412